I cannot tell what you are asking, really. WHat role do $N$ and $N'$ play in all this?
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Mariano Suárez-Alvarez♦Feb 7 '13 at 21:12

@MarianoSuárez-Alvarez : i will do some edit .
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TheoremFeb 7 '13 at 21:15

Your edit did not help much... My mind reading maching suggests the following, though: if you know that $M=A\oplus B$, in particular you know that $M=A+B$, simply by the definition of what a direct sum is.
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Mariano Suárez-Alvarez♦Feb 7 '13 at 21:27

@MarianoSuárez-Alvarez : that means the direct sum and sum are not different at all .
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TheoremFeb 7 '13 at 21:30

No, it does not mean that. As I wrote, if you know that $M=A\oplus B$ then you know that $M=A+B$. I did not say the converse implication holds (and it doesn't)
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Mariano Suárez-Alvarez♦Feb 7 '13 at 21:38

There are a few equivalent ways to define a semisimple module and it would be helpful to know exactly which formulation you are using. One possible formulation is the following:

Definition.A module $M$ is semisimple if every submodule of $M$ is a direct summand of $M$.

Then to prove that every submodule of $M$ is semisimple let $N \subseteq M$ be a submodule and assume $P \subseteq N$ is a submodule of $N$. Write $M = N \oplus N' = P \oplus P'$ for some $N'$ and $P'$. What we need is to find $Q$ such that $N = P \oplus Q$.

Yes , this is more or less what i am following . My question is when can a person write just 'sum' instead of 'direct sum' ?
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TheoremFeb 7 '13 at 21:35

When $A$ and $B$ are submodules of the same module $M$.
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JimFeb 7 '13 at 21:42

@Theorem Adding this because I'm speculating Theorem meant something else with that first comment. If $A$ and $B$ are submodules of $M$ and $A\oplus B=M$ is an (internal) direct sum, you can also write $A+B=M$. So, every (internal) direct sum equal to $M$ is also a sum equal to $M$, but the converse is not true. If $C+D=M$, the sum is not direct unless $C\cap D=\{0\}$. The following can be proven: a module is a sum of simple submodules iff it is a direct sum of simple submodules.
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rschwiebFeb 7 '13 at 22:23